You and your best friend Janine decide to play a game. You are in a land of make believe, where you are a function f(t), and she will be a function g(t). In this make-believe land, the two of you are posing as parametric equations, (to keep other equations from interfering). As parametric equations, your joint path is dependent on decisions that each of you make. Janine decides how you will move in the North and South (y-axis) directions, and you control East and West (x-axis). If your identity, f(t) is given by:
f(t)=(1/3)(t^(2)+8)^(3/2)
and Janine's identity, g(t) is given by:
g(t)=4t
How many units of distance do the two of you cover between the Most Holy Point o' Beginnings (t=0), and The Buck Stops Here (t=11)?
f(t)=(1/3)(t^(2)+8)^(3/2)
and Janine's identity, g(t) is given by:
g(t)=4t
How many units of distance do the two of you cover between the Most Holy Point o' Beginnings (t=0), and The Buck Stops Here (t=11)?
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v=sqrt(f'(t)^2 + g'(t)^2)
f'(t)=3/2 * 1/3 * (t^2+8)^(1/2) * 2t = t(t^2+8)^1/2
g'(t)=4
v=sqrt(t^2(t^2+8) + 16)=sqrt((t^2+4)(t^2+4))=t^2+4
distance travelled = integral (t=0 to t=11) t^2+4 dt
=(t^3/3 + 4t) at t=11
=1331/3 + 44 = 487 and 2/3
f'(t)=3/2 * 1/3 * (t^2+8)^(1/2) * 2t = t(t^2+8)^1/2
g'(t)=4
v=sqrt(t^2(t^2+8) + 16)=sqrt((t^2+4)(t^2+4))=t^2+4
distance travelled = integral (t=0 to t=11) t^2+4 dt
=(t^3/3 + 4t) at t=11
=1331/3 + 44 = 487 and 2/3